##
Steady-state bifurcation with Euclidean symmetry

* Trans. Amer. Math. Soc. * **351** (1999) 1575-1603.

** Ian Melbourne **

** Abstract **

We consider systems of partial differential equations equivariant under
the Euclidean group **E**(n) and undergoing steady-state bifurcation
(with nonzero critical wavenumber) from a fully symmetric equilibrium.
A rigorous reduction procedure is presented that leads
locally to an optimally small system of equations.
In particular, when n=1 and n=2 and for reaction-diffusion equations
with general n, reduction leads to a single equation.
(Our results are valid generically, with perturbations
consisting of relatively bounded partial differential operators.)

In analogy with equivariant bifurcation theory for compact groups, we
give a classification of the different types of reduced systems in
terms of the absolutely irreducible unitary representations of **E**(n).
The representation theory of **E**(n)$ is driven by the irreducible
representations of **O**(n-1). For n=1, this constitutes a
mathematical statement of the `universality' of the Ginzburg-Landau equation
on the line. (In
** recent work **
we addressed the validity of this
equation using related techniques.)

When n=2, there are precisely two
significantly different types of reduced equation:
*scalar* and
**
***pseudoscalar* , corresponding to the trivial and
nontrivial one-dimensional representations of **O**(1).
There are infinitely many possibilities for each n>2.

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